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abaqus - ball indentation- high value of stress

Dear all,

I have modeled a cyclic ball indentation problem in
Abaqus. The sphere has been modeled as axisymmetric rigid surface while
the plate is axisymmetric deformable. The material data fed in for the
plate has been obtained from low cycle fatigue experiments and the
combined hardening model using half-cycle has beem employed.

Apart from this, the loading is displacement-controlled and sphere
is the master surface while the plate is slave surface. The other
parameters used are: surface to surface contact, small sliding, "Hard"
contact and friction coefficient as 0.2.

The meshing is done as is done is various published papers.

PROBLEM: I am getting Stress values (in the vertical direction: S22) as
very high. I am getting values as high as 1600 MPa underneath the
indenter, when the UTS of the material is only 540 MPa.
What could be the reason for this?

It would be kind of you to reply to me as soon as possible.

Thank You,

Aneesh Bangia


Mike Ciavarella's picture

Dear Aneesh


  I find this problem also in my students.

They think that FE commercial softwares can solve everything on earth, and they start from the final complex problem.

I am surprised you didn't use also temperature-dependence of plasticity modulus etc.

Going to the point.  Let me see what you get in simplified cases:

1) cyclic indentation but elastic, and compare to Hertz analytical solution

2) cyclic loading of your material, but in a simple beam -- and compare to analytical solution

3) as 2, but concentrated force in halfspace ---and compare to analytical solution

4) remove friction, add friction

5) increase step of load, decrease step

6) try some of the ABAQUS demo first

I can think of another 50 experiments.  But try these 5 first.

Hope this helps

Michele Ciavarella,
Editor, Italian Science Debate,
YouTube Channel

Dear Aneesh,

The high stresses you are seeing may actually be correct due to the generation of a large hydrostatic stress under the indenter, which can lead to axial stresses that are approximately 3 times the yield strength of the material.


Mike Ciavarella's picture

Chad is rigth, and indeed the Brinell hardness test is  based on that equation.

Check   The BHN can be converted into the ultimate tensile strength
(UTS), although the relationship is dependent on the material, and
therefore determined empirically. The relationship is based on Meyer's
index (n) from Meyer's law. If Meyer's index is less than 2.2 then the ratio of UTS to BHN is 0.36. If Meyer's index is greater then the ratio increases.[1]

Since the hydrostatic stress depends largely on Poisson's ratio, try to change that.

You should not be sad however, since Heinrich Hertz himself made
this mistake you are making, as he assumed the plastic flow would start
from the surface in his third paper on the subject when he firstly proposed to define hardness from the local pressure on surface.


Contact mechanics

Memorial of Heinrich Hertz on the campus of the Karlsruhe Institute of Technology

Main article: Contact mechanics

In 1881–1882, Hertz published two articles on what was to become known as the field of contact mechanics. Hertz is well known for his contributions to the field of electrodynamics (see below);
however, most papers that look into the fundamental nature of contact
cite his two papers as a source for some important ideas. Joseph Valentin Boussinesq
published some critically important observations on Hertz's work,
nevertheless establishing this work on contact mechanics to be of
immense importance. His work basically summarises how two axi-symmetric
objects placed in contact will behave under loading, he obtained
results based upon the classical theory of elasticity and continuum
mechanics. The most significant failure of his theory was the neglect
of any nature of adhesion between the two solids, which proves to be
important as the materials composing the solids start to assume high
elasticity. It was natural to neglect adhesion in that age as there
were no experimental methods of testing for it.

To develop his theory Hertz used his observation of elliptical Newton's rings
formed upon placing a glass sphere upon a lens as the basis of assuming
that the pressure exerted by the sphere follows an elliptical
distribution. He used the formation of Newton's rings again while
validating his theory with experiments in calculating the displacement
which the sphere has into the lens. K. L. Johnson, K. Kendall and A. D.
Roberts (JKR) used this theory as a basis while calculating the
theoretical displacement or indentation depth in the presence
of adhesion in their landmark article "Surface energy and contact of
elastic solids" published in 1971 in the Proceedings of the Royal
Society (A324, 1558, 301-313). Hertz's theory is recovered from their
formulation if the adhesion of the materials is assumed to be zero.
Similar to this theory, however using different assumptions, B. V. Derjaguin,
V. M. Muller and Y. P. Toporov published another theory in 1975, which
came to be known as the DMT theory in the research community, which
also recovered Hertz's formulations under the assumption of zero
adhesion. This DMT theory proved to be rather premature and needed
several revisions before it came to be accepted as another material
contact theory in addition to the JKR theory. Both the DMT and the JKR
theories form the basis of contact mechanics upon which all transition
contact models are based and used in material parameter prediction in
Nanoindentation and Atomic Force Microscopy. So Hertz's research from
his days as a lecturer, preceding his great work on electromagnetism,
which he himself considered with his characteristic soberness to be
trivial, has come down to the age of nanotechnology.


You may also like to visit Wikipedia


Contact mechanics
From Wikipedia, the free encyclopedia

Jump to: navigation, search

Continuum mechanics




Conservation of mass
Conservation of momentum
Conservation of energy
Entropy inequality

[show]Solid mechanics

Solids · Stress · Deformation · Finite strain theory · Infinitesimal strain theory · Elasticity · Linear elasticity · Plasticity · Viscoelasticity · Hooke's law · Rheology

[show]Fluid mechanics

Fluids · Fluid statics
Fluid dynamics · Viscosity · Newtonian fluids
Non-Newtonian fluids
Surface tension


Newton · Stokes · Navier · Cauchy · Hooke · Bernoulli

This box: view  talk  edit

Stresses in a contact area loaded simultaneously with a normal and a
tangential force. Stresses were made visible using photoelasticity.

Contact mechanics is the study of the deformation of solids that touch each other at one or more points[1][2]. The physical and mathematical formulation of the subject is built upon the mechanics of materials and continuum mechanics and focuses on computations involving elastic, viscoelastic, and plastic bodies in static or dynamic contact. Contact mechanics is foundational to the field of mechanical engineering; it provides necessary information for the safe and energy efficient design of technical systems.

The original work in contact mechanics dates back to 1882 with the publication of the paper "On the contact of elastic solids" ("Ueber die Berührung fester elastischer Körper") by Heinrich Hertz. Hertz was attempting to understand how the optical properties of multiple, stacked lenses might change with the force
holding them together. Results in this field have since been extended
to all branches of engineering, but are most essential in the study of tribology and indentation hardness.
Hertzian contact stress refers to the localized stresses that develop
as two curved surfaces come in contact and deform slightly under the
imposed loads. This amount of deformation is dependent on the modulus of elasticity
of the material in contact. It gives the contact stress as a function
of the normal contact force, the radii of curvature of both bodies and
the modulus of elasticity of both bodies. In gears and bearings in
operation, these contact stresses are cyclic in nature and over time
lead to sub-surface fatigue cracks. Hertzian contact stress forms the
foundation for the equations for load bearing capabilities in bearings,
gears, and any other bodies where two surfaces are in contact.

Principles of contacts mechanics can be applied in areas such as locomotive wheel-rail contact, coupling devices, braking systems, tires, bearings, combustion engines, mechanical linkages, gasket seals, metalworking, metal forming, ultrasonic welding, electrical contacts, and many others. Current challenges faced in the field may include stress analysis of contact and coupling members and the influence of lubrication and material design on friction and wear. Applications of contact mechanics further extend into the micro- and nanotechnological realm.

The motion of a single body in space is described by the governing equations of continuum mechanics.
The approach used in contact mechanics is to restrict the the motion of
two or more bodies in space by additional constraints. These unilateral
constraints ensure that bodies do not penetrate each other after coming
into contact. Once the general equations for a contact problem are set
up, different solution schemes can be used to simulate the behaviour of
bodies in contact and to compute displacement and stress fields. A
distinction is usually drawn between contact with and without friction.



Michele Ciavarella,
Editor, Italian Science Debate,
YouTube Channel

That was a relief, really.

thanks a lot Mike and Chad!

i cannot however relate BHN to UTS because the material parameter has to be obtained emperically.

In order to validate my results from simulation, I compared my Force vs Displacement curves with that obtained experimentally with the ones obtained by simulation.

The force value obtained through simulation was found to be about 1.5 times that of the value obtained through indentation experiments.( 2080 N compared to 1400 N)

This is one benchmark I need to confirm what I'm doing is correct.

Also, it must be pointed out here that the experiments were performed sometime back by some other student, and I am just using his experimental results. 

Mike Ciavarella's picture



Frankly, now again I don't undertstand!  Why do you beleive your LCF data are absolutely good, to be beleived as GOD?  Actually, on the contrary, the more DISTANT you are from the conditions you simulate in FEM tests, the more OBVIOUS is that you do not fit correctly !

To relate hardness to UTS, what do you mean you cannot do it?

Here, it is ALL empirical fitting.  Once you decide what to best fit, the rest is not exact, but approximate. 

Ideally, the easiest to fit is the test itself!  In other words, you generate a plastic material that is generating that indentation test.  Of course when you use back into LCF fatigue data, you are lost again.... ;)

So please reformulate.  As I told you, to relate indentation laws with Meyer's "law"  --- the term "law" as usual is misused by engineers as if it were a law of physics.   In fact, since it is quite general, you can fit almost ANYTHING to it --- this is perhaps to say this is really not a law, or else that this is the common mistake with power laws, unfortunately these fitting equations are considered "laws"!

Perhaps your assignement is not well posed.  I additionally do not understand if you want to do cyclic indentation fit testing with LCF data or not.  It reminds me of my work in Rolling Contact Fatigue, where contact stresses have been attempted to be related to LCF plasticity "laws" for ages, without much success -- please read these two papers of mine:

A re-examination of rolling contact fatigue
experiments by Clayton and Su with suggestions for surface durability

Wear, Volume 256, Issues 3-4, February
, Pages 329-334
L. Afferrante, M. Ciavarella, G. Demelio

A note on Merwin’s measurements of forward flow in
rolling contact

Wear, Volume 256, Issues 3-4,
February 2004, Pages 321-328
A. R. S. Ponter, L.
Afferrante, M. Ciavarella


 And read again WIKIPEDIA please.

Meyer's law
From Wikipedia, the free encyclopedia

Jump to: navigation,

Meyer's law is an empirical
relation between the size of a hardness test indentation
and the load required to leave the indentation.[1]


[edit] Equation

It take the form:



  • P = pressure in megapascals
  • k = property of the material
  • n = Meyer's index, a property of the material
  • d = chordal diameter (diameter of the indentation)

n usually lies between the values of 2, for fully strain hardened materials, and 2.5,
for fully annealed materials. It is roughly related to the
strain hardening coefficient in the equation for the true stress-true
strain curve by adding 2.[1]
Note, however, that below approximately d = 0.5 mm (0.020 in) the value
of n can surpass 3. Because of this Meyer's law is often restricted to
values of d greater than 0.5 mm up to the diameter of the indenter.[2]

The variables k and n are also dependent on the size of the indenter.
Despite this, it has been found that the values can be related using
the equation:[3]

P = k_1d_1^{n_1} = k_2d_2^{n_2} = k_3d_3^{n_3} =<br />

Meyer's law is often used to relate hardness values based on the fact
that if the weight is halved and the diameter of the indenter is
quartered. For instance, the hardness value for a test load of 3000 kg
and a 10 mm indenter is the same for a test load of 750 kg and a 5 mm
diameter indenter. This relationship isn't perfect, but its percent error is relatively small.[4]

A modified form of this equation was put forth by Onitsch:[5]


[edit] See also

[edit] References
[edit] Notes

  1. ^ a
    Hardness Testing,, retrieved 2008-10-07 .
  2. ^
    Tabor, pp. 12-14.
  3. ^
    Tabor, p. 8.
  4. ^
    Tabor, pp. 10-11.
  5. ^ Blau,
    P. J.; Lawn, Brian R.; American Society for Testing and Materials
    Committee E-4 on Metallography, International Metallographic Society
    (1986), Microindentation Techniques in
    Materials Science and Engineering
    , ASTM International, p. 93, ISBN 0803104413,

[edit] Bibliography



Michele Ciavarella,
Editor, Italian Science Debate,
YouTube Channel

Hi Mike,

Thanks for the direction, again.

I found out the Meyer's index which was below 2.2; and my UTS was 560.5
MPa and BHN came out to be 1569.3 (in MPa). UTS/BHN = 0.357

(which is close to 0.36, as written in wiki)

So, does this kind of validate the high stresses observed underneath the indenter?

I would now like to shed some light on what my project is about and what all I have done already.

I have to correlate the plastic dissipation energies for LCF and
cyclic indentation. For that the experiemental part has already been
performed by another student, and I am doing the simulation part. I
have already performed simulations for LCF and obtained good results
(hysteresis curves, plastic dissipation energy curves, and so on)

But, I've been facing problems in the case of indentation. Once I'm
sure that the stress values that I've obtained are actually correct and
other things such as Force vs displacement curves are matching with
indentation experimental curves to an extent (with little error), I can
go ahead with correlating the plastic dissipation energy for the two

Meanwhile, I am reading the papers as suggested by you.

Thanks and Regards,


Mike Ciavarella's picture

Modeling of the cyclic ball indentation test for small
specimens using the finite …

T Yamamoto, H Kurishita, H Matsui - Journal of Nuclear
Materials, 1999 - Elsevier

... Fig. 6 shows a typical
load-displacement curve from the cyclic indentation test
conducted for
the as-machined specimen together with the FEM simulation of the tests.
The experimental results
showed much larger elastic deformations than any simulation result. ...


Cyclic indentation in aluminum

F Yang, L Peng, K Okazaki - Journal of Materials Science,
2007 - Springer

... Al F, d Fig. 1 Schematic of the cyclic
indentation test of Al 123 ... Page 3. Results and
Indentation fatigue curves During the cyclic indentation test,
a cyclic indentation load was applied
to the indenter, which caused the indenter to move into the surface of
the specimen. ...

of Material Fatigue Behavior Through Cyclic Ball Indentation Testing

RV Prakash, P Bhokardole, CS Shin -
Journal of ASTM International, 2008 -

... The
purpose of this work is to examine the fatigue behavior of materials
through an in-situ test
technique, viz., cyclic indentation test method. ...
This record corresponds to a cyclic indentation
test that was carried out between −50 N and −500 N force at a
frequency of 0.25 Hz. ...


Numerical investigation of indentation fatigue on
polycrystalline copper

Xu, ZF Yue, X Chen - J. Mater. Res, 2009 -

loop in the indentation load–depth curve that depends on the dopant
level and the depth of
indentation.13 This study may shed some light on extracting sensitive
material information such
as the sen- sitive defect–solute interactions by using the cyclic
indentation test, which are ...

- ACNP Posseduto Biblioteche - Tutte
e 4 le versioni

indentation-induced deformation microstructures in GaN thin films

CH Chien, SR Jian, CT Wang, JY Juang,
JC … - Journal of Physics D: …, 2007 -

previous load, which completed the first cycle. It then was reloaded to a
larger chosen
load and unloaded by 90% for the second cycle. Figure 1 illustrated a
typical cyclic
indentation test repeated for 5 cycles. It is noted that
in each ...

- ACNP Posseduto Biblioteche

An experimental methodology for characterizing fracture of
hard thin films under …

Yonezu, B Xu, X Chen - Thin Solid Films, 2009 - Elsevier

Unfortunately, such signal may be difficult to detect during cyclic
indentation test. ... Fig.
1b shows the conventional cyclic indentation test
with a constant force range ΔF.
Detailed test conditions are described in the following sections. ...

da 1
- Articoli
- Tutte
e 2 le versioni

[PDF] Application of the indentation method for cracking
resistance evaluation of … [PDF]
A Sivitski, A Gregor, M Saarna, P
Kulu, F … - Estonian Journal of …, 2009 -

475 6 200 – AlTiN –60…–150 4 × 10–3…1.2 × 10–2 (60…125)/(52...130)
430…450 6 150…200 –
2.3. Cyclic indentation test procedure A servo
hydraulic fatigue test system INSTRON 8800 and
Vickers diamond pyramid indenter were used in the indentation
experiments. ...


An indentation system for determination of viscoplastic
stress-strain behavior of … [PDF]
N Huber, E Tyulyukovskiy, HC
Schneider, R … - Journal of Nuclear …, 2008 - Elsevier

development of fusion materials for the first wall in future fusion
reactors requires methods
for the investigation of irradiation effects on the mechanical
properties of materials which are
only available in small volumes. Depth and force reading hardness
measurement (or ...
da 6
- Articoli
- ACNP Posseduto Biblioteche - Tutte
e 8 le versioni

Study of ratcheting by the indentation fatigue method with a
flat cylindrical indenter. …

BX Xu, ZF Yue - Journal of Materials Research, 2007 -

The finite element method (FEM) was used to study the
flat cylindrical indentation fatigue behavior
using a kinematic hardening model (AF model). This study was motivated
by the experimental
work of the preceding paper [BX Xu and ZF Yue, J. Mater. Res. 21, 1793
(2006)], in which ...
da 5
- Articoli
- ACNP Posseduto Biblioteche - Tutte
e 8 le versioni

A study on determining hardening curve for sheet metal
H Tian, D Kang - International Journal
of Machine Tools and …, 2003 - Elsevier

The hardening curve
for sheet metal can be determined from the load–displacement curve of
tensile specimen with rectangular cross-section. The previous
researches, however, have paid
little attention to its use in large deformation. Moreover, it varies
with materials, ...
da 2
- Articoli
- ACNP Posseduto Biblioteche - Tutte
e 6 le versioni

SS Akarca, WJ Altenhof, AT Alpas -

S. Subutay Akarca, University of Windsor 401
Sunset Ave. Windsor, ON, CANADA N9B 3P4
Telephone: (519) 253 3000 ext.2605 Facsimile: (519) 973 7007 ...
Dr. William J. Altenhof, University
of Windsor 401 Sunset Ave. Windsor, ON, CANADA N9B 3P4 Telephone: (519)
253 3000 ...


You probably have them all, but just to make sure  you have !!




Michele Ciavarella,
Editor, Italian Science Debate,
YouTube Channel

those links were very helpful Mike, especially the first two.


 Now that the problem of high stress has been resolved, comes the issue of Plastic Dissipation energy.

As i have said before, my goal is to correlate the plastic dissipation energy for Low Cycle Fatigue with Cyclic Indentation.

The doubt right now is as follows:

I am getting a huge difference in magnitude of PDE for Low Cyc Fatigue and CI. For one cycle, PDE for LCF is around 35 J while it is only 7 mJ for indentation. That is, about 5000 times greater.

Does this seem correct to you?

Mike Ciavarella's picture


You are not very kind.  You say "they were very useful"... in what?  Here, I am willing to help, for curiosity, but in return, I expect my curiosity to be satisfied!

So please provide full answer, I am not a cheap "vendor machine" of technical suggestions, I was expecting a civilized technical discussion between peers :)

For the next question, how do you compute PDE in the two cases?


Michele Ciavarella,
Editor, Italian Science Debate,
YouTube Channel

sorry Mike for being brusque.

The first paper was very useful because it clearly showed the
contours of Principal Stress whose values are shooting up in the range
that I am getting. (Although the material used was different from mine, these values were much greater than its Yield Strength). This vindicates my results and I am satisfied. Also,
it tells me the interpretation of the positive values of these stresses
and that these can be compared with the yield stress. The Force vs
Displacement curve's pattern matches with mine but i could get much out
of it.

The second paper was useful because it helped me in
improving my theoretical knowledge about Plastic Dissipation energy,
which is my prime area of concern. It also gave me a good idea about
how a load-controlled load should be provided, which is the case I may
examine next. The paper also plots PDE vs Force with PDE in micro-Joule
while Force in mN. So, I thinks it means that the order PDE I am
getting (in mJ for force in N) could be correct.

Also, Mike I
must point out that one of the authors is my faculty guide and the
other is whose experimental work I am continuing.

For the
papers, I must frankly say that I have only had a cursory glance
through them so far. But I look forward to go through them in detail in
the coming days. The list was exhaustive and I thank you Mike for
making the effort for me.

I am now pretty confident about the Stress vs Strain curves and the Force-displacement curves that I am getting because I've a good explaination for them.

Now coming to your question about PDE, I obtained it through the 'History Output Request' in ABAQUS.

I must say that I am really happy to include you in my progress of the project.



Mike Ciavarella's picture

All is well then.  However, next time read your supervisor's papers, before asking imechanica!!  I know it is nicer to find someone willing to help like me, and I am learning also in the process, but it sounds funny this very modern complex way to do the obvious thing to do!

Most supervisors ask for project works and thesis, continuation of work of previous students...

On the other hand, sometimes it is BETTER not to read the previous work, so that you can come out with better ideas, or new ideas.  So don't follow your supervisor too much!    This is why, for example, they say Einstein had those brilliant ideas at age 16-26 only because he was NOT working in academia, writing proposals, following supervisor's ideas :)  Who would ever ask for phd to devise a relativity theory????



Michele Ciavarella,
Editor, Italian Science Debate,
YouTube Channel

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